Due to the substantial similarities between the two algorithms, we refer to both
as regret matching. Both forms of regret matching have been shown to be special cases of a general class of potential-based minimizers [1]}, [2]}, [3]}. Specifically, many of their theoretical properties can be proved via careful examination of the potential function, defined as the sum of the squared positive regrets. Formally, the potential functions that regret matching minimizes for external and internal regret, respectively, are:
\(\phi ({\bf \bar{R}}_+^{E,T}) = \sum _{i \in P} \sum _{a_i \in A_i} \big (\bar{R}_{i,+}^{E,T}(a_i)\big )^2\) 
\(\phi ({\bf \bar{R}}_+^{I,T}) = \sum _{i \in P} \sum _{a^A_i, a^B_i \in A_i} \big (\bar{R}^{I,T}_{i,+}(a^A_i, a^B_i)\big )^2\) 
